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    Riemann's removable singularity theorem

    T-ZZJDP

    Removability is exactly the absence of a principal part, equivalently boundedness or holomorphic extension across the point.Suggest a slogan

    Let \(\Omega\subseteq\CC\) be open, let \(z_0\in\Omega\), and let \(f\) be holomorphic on \(\Omega\setminus\{z_0\}\). The following are equivalent.

    1. \(z_0\) is a removable singularity of \(f\).

    2. \(f\) is bounded on \(\{z : 0<\abs{z-z_0}<\delta\}\) for some \(\delta>0\).

    3. \((z-z_0)f(z)\to0\) as \(z\to z_0\).

    4. \(f\) extends to a holomorphic function on \(\Omega\).

    5. \(f\) extends to a continuous function on \(\Omega\).

    6. The Laurent expansion of \(f\) on a punctured disc \(\{z : 0<\abs{z-z_0}<\delta\}\subseteq\Omega\) has vanishing principal part: \(f(z)=\sum_{k\geq0}c_k(z-z_0)^k\) there.

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    • T-LZ6KGExtension over removable singularitiesvariant-of
    • T-KOZO4Riemann extension theoremvariant-of
    • T-AL6OTRiemann's removable singularity theoremvariant-of

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